Question

Which of the following is a sum of cubes?

A)3x^3+8y^6
B)8x^3-27y^6
C)125x^6-9y^3
D)x^3+8y^6

Answer 1

Hello!

The answer is:

D)
`x^(3)+8y^(6)`

Why?

We are looking for the expression that can be obtained from a sum of cube factoring, the form of the sum of cubes will be:

`a^(3)+b^(3)=(a+b)(a^(2) -ab+b^(2))`

The only option that matches with the sum of cubes form is the last option,

D)
`x^(3)+8y^(6)`

and it can be rewritten as the following expression:

`x^(3)+8y^(6)=x^(3)+(2y^(2))^(3)\\\\x^(3)+(2y^(2))^(3)=(x+2y^(2))(x^(2)-2xy^(2)+(2y^(2))^(2))\\\\(x+2y^(2))(x^(2)-2xy^(2)+(2y^(2))^(2))=(x+2y^(2))(x^(2)-2xy^(2)+4y^(4))\\\\(x+2y^(2))(x^(2)-xy^(2)+4y^(4))=x*x^(2)-2x^(2)y^(2)+2xy^(4)+2x^(2)y^(2)-2xy^(4)+8y^(6)\\\\x*x^(2)-2x^(2)y^(2)+2xy^(4)+2x^(2)y^(2)-2xy^(4)+8y^(6)=x^(3)+8y^(6)`

Therefore, we have that:

`x^(3)+8y^(6)=x^(3)+(2y^(2))^(3)=(x+2y^(2))(x^(2)-2xy^(2)+(2y^(2))^(2))=x^(3)+8y^(6)`

Hence, the correct answer is:

D)
`x^(3)+8y^(6)`

Have a nice day!

Answer 2

D)
`x^3+8y^6`

Explanation:

We have been given four choices.

A)
`3x^3+8y^6`

B)
`8x^3-27y^6`

C)
`125x^6-9y^3`

D)
`x^3+8y^6`

Now we need to find about which of the above choices is a sum of cubes.

Basically we need to check which one of the given choices can be represented in cubic form along with addition sign.

3 and 9 can't be written in cubic form unless we use radical numbers.

So the only choice left is D)

D)
`x^3+8y^6`

`=x^3+2^3(y^2)^3`

`=x^3+(2y^2)^3`

which is clearly visible as sum of cubes.